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Factorize the...

Question

Factorize the following:
$5 - (3 a^{2} - 2 a) (6 - 3 a^{2} + 2 a)$

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Solution

The given expression $5 - (3 a^{2} - 2 a) (6 - 3 a^{2} + 2 a)$ can be factorized as follows:

$5 - (3 a^{2} - 2 a) (6 - 3 a^{2} + 2 a) = 5 - (3 a^{2} - 2 a) [6 - (3 a^{2} - 2 a)] = 5 - y (6 - y) (A s s u m i n g 3 a^{2} - 2 a = y) = 5 - 6 y + y^{2} = y^{2} - 6 y + 5 = y^{2} - 5 y - y + 5 = y (y - 5) - 1 (y - 5) = (y - 5) (y - 1) = (3 a^{2} - 2 a - 5) (3 a^{2} - 2 a - 1) (∵ 3 a^{2} - 2 a = y) = (3 a^{2} + 3 a - 5 a - 5) (3 a^{2} - 3 a + a - 1) = [3 a (a + 1) - 5 (a + 1)] [3 a (a - 1) + 1 (a - 1)] = (3 a - 5) (a - 1) (a + 1) (3 a + 1)$

Hence, the factorization of $5 - (3 a^{2} - 2 a) (6 - 3 a^{2} + 2 a)$ is $(3 a - 5) (a - 1) (a + 1) (3 a + 1)$ .

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